[Paper Review] A counterexample to the reconstruction of $\omega$-categorical structures from their endomorphism monoids

Manuel Bodirsky, David M. Evans+2 more|arXiv (Cornell University)|Oct 1, 2015

Advanced Algebra and Logic9 references7 citations

TL;DR

This paper presents a counterexample to the reconstruction of ω-categorical structures from their endomorphism monoids, showing that two countable ω-categorical structures in finite relational languages can have isomorphic endomorphism monoids that are not topologically isomorphic. The key contribution is proving that such structures cannot be reconstructed up to existential positive bi-interpretations from their endomorphism monoids, nor up to first-order bi-interpretations from their automorphism groups, nor up to primitive positive bi-interpretations from their polymorphism clones.

ABSTRACT

We present an example of two countable $\omega$-categorical structures, one of which has a finite relational language, whose endomorphism monoids are isomorphic as abstract monoids, but not as topological monoids -- in other words, no isomorphism between these monoids is a homeomorphism. For the same two structures, the automorphism groups and polymorphism clones are isomorphic, but not topologically isomorphic. In particular, there exists a countable $\omega$-categorical structure in a finite relational language which can neither be reconstructed up to first-order bi-interpretations from its automorphism group, nor up to existential positive bi-interpretations from its endomorphism monoid, nor up to primitive positive bi-interpretations from its polymorphism clone.

Motivation & Objective

• To investigate whether ω-categorical structures can be reconstructed from their automorphism groups, endomorphism monoids, or polymorphism clones.
• To determine whether isomorphisms between these algebraic objects imply topological isomorphisms.
• To resolve open questions about the consistency of reconstruction for monoids and groups in the context of ω-categorical structures.
• To construct explicit examples of ω-categorical structures where reconstruction fails even under weak notions of interpretability.

Proposed method

• Leveraging a construction from Evans and Hewitt (1990) involving non-topological isomorphisms between oligomorphic groups.
• Using Hrushovski's method to encode structures with infinite relational languages into finite relational languages while preserving model-completeness.
• Constructing a structure C with finite relational signature such that its endomorphism monoid is isomorphic but not topologically isomorphic to that of a base structure B.
• Proving that the polymorphism clone of C is isomorphic to the function clone generated by the automorphism group of a related structure, but not topologically isomorphic.
• Employing topological group theory and the closure of oligomorphic groups in ωω to analyze monoid and group isomorphisms.
• Using model completeness and homogeneity to ensure that the constructed structures lie in the age of the target structure and satisfy required relations.

Experimental results

Research questions

• RQ1 Can an ω-categorical structure be reconstructed up to first-order bi-interpretation from its automorphism group?
• RQ2 Does an isomorphism between endomorphism monoids of ω-categorical structures imply a topological isomorphism?
• RQ3 Can polymorphism clones of ω-categorical structures fail to be reconstructible even when isomorphic as abstract clones?
• RQ4 Is there a finite relational language ω-categorical structure whose endomorphism monoid does not allow reconstruction via topological isomorphism?
• RQ5 Does the closure of a non-reconstructible oligomorphic group in ωω also fail to have reconstruction?

Key findings

• There exists a countable ω-categorical structure A in a finite relational language such that its endomorphism monoid is isomorphic but not topologically isomorphic to that of another structure B.
• The automorphism group of A is not topologically isomorphic to that of B, despite being abstractly isomorphic, showing that reconstruction fails even at the group level.
• The polymorphism clone of A is isomorphic but not topologically isomorphic to that of B, demonstrating failure of reconstruction for clones.
• The example is constructed via Hrushovski's encoding, preserving model-completeness and ensuring finite relational signature.
• The monoid closure of an oligomorphic group that lacks reconstruction also fails to have reconstruction, extending the counterexample to monoids.
• The result provides a negative answer to the question of whether isomorphisms between endomorphism monoids of ω-categorical structures must be topological, even in finite relational languages.